1 DEFINITION OF A MEASURE 99 and the properties of measures defined by densities (13.15) give us a method for dealing with denumerable families of measures on a locally compact space, by fixing a basic measure and working with the densities relative to this basic measure (hence again with functions); this again proves to be extremely convenient. Here the modern point of view emerges: given a /^-integrable function/, what is important is not the values taken by/so much as the way in which /operates on the space of bounded continuous functions by means of the linear mapping #h-» \fg d\i (this mapping depends only on the equiva- lence class of/and therefore does not change when we modify/at the points of a set of measure zero). The development of this point of view will lead in Chapter XVII to the theory of distributions, which is a natural generalization of the notion of measure on differential manifolds. Throughout this chapter, the phrase "locally compact space" will always mean "separable metrizable locally compact space." 1. DEFINITION OF A MEASURE To begin with, let X be a compact (metrizable) space. A measure (or complex measure) on X is by definition an element of the dual of the Banach space ^cW of complex-valued continuous functions on X (7.2), that is to say ((12.15) and (5.5.1)) it is a linear form /h-> /*(/) on ^C(X) which satisfies an inequality of the form (13.1.1) for all/e ^C(X) (recall that ||/|| = sup xeX Now let X be a locally compact space (metrizable and separable, in accordance with our conventions). For every compact subset K of X, let JT(X; K) (or Jf*c(X; K)) denote the vector subspace of ^C(X) consisting of the functions whose support (12.6) is contained in K (and is therefore compact). We shall denote by Jfc(X) (or Jf(X)) the union of the Jfc(X; K) as K runs through all compact subsets of X. In other words, JfcW is the vector space of (complex-valued) continuous functions with compact support. Clearly JTC(X) c «£(X). A measure (or complex measure) on X is by definition a linear form \JL on «?TC(X) with the following property: for each compact subset K of X, there exists a real number aK ^ 0 (in general depending on K) such that (13.12) forall/ejf(X;K). J r /